1.06    Combining Like Terms;

                          The Distributive Property

 from Basic Algebra: One Step at a Time © 2002

P. 31-38

In the last section, the concept of variables was introduced.  Variables usually represent either known or unknown numbers.  If the values of the variables are given as they were in the last section, then you can substitute those values in place of the variables and obtain a numerical answer.  When the values of the variables are not known, the expressions may sometimes be simplified by a process called combining like terms.  Before beginning this, however, a few preliminary remarks and definitions may be helpful.

1.      TERMS are those quantities which are added (or subtracted) together.  FACTORS are quantities that are multiplied.  In the expression “x + 5," the x and the 5 are terms, since they are added.  In the expression “5x,” the x and the 5 are factors, since they are multiplied.  In the expression “5x + 3," the 5x and the 3 are terms, but the 5 and the x could be considered to be factors of the 5x term.

2.       LIKE TERMS, or similar terms, are terms which have the same letters (variables) raised to the same power.  Like terms and only like terms may be combined.  For example, the expression 3x + 4x contains x terms.  In the same way that 3 apples plus 4 apples would be 7 apples, the x terms can be combined: 3x + 4x = 7x.  As another example, what about the expression 3x² + 4x² ?  When you add apples to apples, you get apples.  When you add x’s to x’s, you get x’s.  So, it should be no surprise to learn that when you add x²’s to x²’s, you get x²’s.  Therefore, 3x² + 4x² = 7x².

3.       UNLIKE TERMS are those terms which do not have the same letters and powers.  Unlike terms cannot be combined. 

            Examples:       a) 3x + 4y;       b)   3x² + 4x;       c)   3x ² + 4y²

            These examples cannot be simplified, since in each example there are no like terms.

4.        A term may contain more than one variable: 7xy + 3xy = _________ (Answer: 10xy)

5.      A term may have no variable.  We call this a number term.  Number terms can be combined only with other number terms.  4x + 3 + 7y + 6 = __________   

             (Answer: 4x + 7y + 9)

6.     It makes no difference in what order the terms are given.  “4x + 7y + 9" could also be written as “9 + 4x + 7y” or “7y + 9 + 4x” or any other order.  Usually we put the terms in alphabetical order.  When powers are used, such as x² + 3x + 5, again, order does not matter, but usually the highest powers of the variable are placed first and the number term given last.  This is called descending powers of the variable.

7.        A variable alone, such as x, really means 1x or 1 x.  So, 3y + y means the same as 3y + 1y, which is 4y.  Likewise, 4y - 3y would be 1y, but it is simpler to write 4y - 3y = y.

8.        Simplifying the expression 4y - 4y, the result is “no y” or “0 y.”  However, since zero times any number is zero, the answer is simply “0.”

Notice that:      4x + 3y - 4x simplifies to 3y               The 0x is not necessary!

                                   8x - 5x - 3x simplifies to  0                 The 0x (which is 0) is necessary!

9.       Exactly what is meant by 3x?  Answer: 3x means 3 times x, or three x’s added together (x + x + x).  In this case, the number 3 is called the coefficient of x.

10.      What is meant by 3x², and is it the same as (3x)²?     Because there are no parentheses in the expression 3x², according to the order of operations agreement, the x is squared, then the result is multiplied by 3.  Therefore, 3x² means 3 x x.  However, because of the parentheses in (3x)², the quantity to be squared (that is, multiplied times itself!) is 3x.  Therefore, (3x)² means (3x) (3x) or 3 x 3 x.  When multiplying (or adding!) numbers, it can be done in any order.  It is convenient to multiply the 3's together, then the x’s. 

             So,  (3x)²    means    (3x) • (3x)

                                               3x • 3x

                                              3 • 3 •  x • x

                                                 9x² 

Additional example:    5x²  • 6x   equals 30 x³,                    

                  whereas:   (5x)² • 6x means  (5x) (5x) • 6x  or  5• 5 • 6 • x • x • x  or  150 x³.

EXAMPLES:  Simplify by combining all like terms.  Remember, answers may be in any order.

1.  2x + 4x + 3y = _______                                                  ANS:         6x + 3y  

2.  7x + 4y + 3x = _______                                                  ANS:        10x + 4y  

3.  2x + 4 + 3y + x + 10y = __________                              ANS:    3x + 13y + 4   

4.  7x² + 3y² + 3x + 2x² + y² = ______________               ANS:    9x² + 4y² + 3x

5.  12x - 8y + 3y - 8x + x - 7y + 7y² - 8

    = _______________________________                         ANS:   5x + 7y² - 12y - 8

6.  13x - 4y² - 9x + y² + 4x + 3y² - 7x

    = _______________________________                         ANS:   x

7.  12x² - 8y - 9xy + y² + 8x - 3xy² - 8

     = _______________________________                        ANS:   Same as the problem!

                                                                                                           Cannot be simplified!

EXERCISES: Simplify by combining all like terms.

1.  5x - 2x = _______                         2.  5d - 3d = _______

3.  8x + x  = _______                         4.  3b + 3g = _______

 5.   h + h  = _______                         6.  9x + x  = _______

 7.  9c - 4c = _______                         8.  9x - 8x = _______

 9.   x + 3y + 4 = _______                 10.  5w + 6w - 3w = _______

11.  4x - 3y + 12x - 2y = _________________

12.  8k + 7v + 3k - 2v  = __________________

13.  2a + 3c - 8a - 4c  = _________________

14.  12x - 6y + 4z - 3z +11y + x = __________________

15.  xyz + 3xy + 2xyz + xy =   _________________

16.  2LH + 2WH + 2LW - 6WH + 6LH = __________________

17.  7x² + 3x²   = ______________                18.  7y² + 3y² = _____________

19.  7x² + 3y²   = ______________                20.  7y² + 3y = _____________

21.  7x² - 3x²   = ______________                 22. -7y² - 3y² = _____________

23.  3x² + 5x²y² + 9y² = __________________

24.  14x² + 6x  - 8y² - 6xy + 3y² + 12xy = __________________

The Distributive Property

Consider the following examples

EXAMPLE 8.   Find the value of   5(4 + 6)   and    5·4 + 5·6

Solution:                                            5 (10)                20 + 30

                                                            50                       50

EXAMPLE 9.    Find the value of     10(8 + 7)  and   10·8 + 10·7

Solution:                                            10 (15)                 80 + 70

                                                             150                      150

EXAMPLE 10.   Find the value of   10(8 - 7)   and   10·8 - 10·7

Solution:                                               10 (1)                 80 - 70

                                                                10                       10

These are examples of the distributive property, formally called the distributive property for multiplication over addition.  The distributive property is much more than a “neat trick” that works with numbers.  It is one of the most important and frequently used properties in all of math.  You really need to know and be able to use this one, both now and for future reference.  This property is formally stated as follows:

EXAMPLE 11.    6(x + 5)     According to the order of operations agreement, you must perform the addition within the parentheses before you multiply.  However, these are unlike terms, so they cannot be combined.  Therefore, in order to eliminate the parentheses, you can use the distributive property as follows:       

                             6(x + 5) = 6·x + 6·5

                                           =  6x + 30             

The distributive property also works when subtracting and when using longer expressions.

EXAMPLE 12.   6(x - 5) =  6x - 30

EXAMPLE 13.   6(x + 3y - 7) =  6x + 18y - 42

In the next examples, remember that x·x = x²

EXAMPLE 14.    x (x - y + 4) = x² - xy + 4x 

EXAMPLE 15.   7 (3x + 5y - 6) = 21x + 35y - 42  

EXAMPLE 16.  -7x (3x + 5y - 6) = -21x² - 35xy + 42x  

EXERCISES:   Use the distributive property to remove the parentheses.

25.     5 (x + 3) = __________                      26.       5 (x + 5) = __________

27.      6 (x - 4) = __________                      28.       8 (y - 9) = ___________

29.     7 (2x + 5) = __________                    30.       5 (4x + 7) = __________

31.      6 (9x - 8) = __________                    32.       9 (7y - 9) = __________

33.     -5 (x - 5) = __________                      34.      -6 (x + 5) = __________

35.     -1 (3x + 7) = __________                  36.       -1 (3y - 4) = __________

37.     - (-8x - 7) = __________                    38.        - (-8x + 5) = __________

39.      5x (x - 3) = _________________        40.       8y (y + 9) = __________________

41.      8x (5x + 3y) = _______________        42.      5x (7x - 5y) = ________________

43.     5x (2y-3x+7) = ______________          44.      4y (7x-9y-6) = ________________

45.    -6y (2y+3x+7) = _____________           46.     -3x (7x-5y+1) = _______________

47.    -7x (5y-x+1) = _______________          48.     -4y (-7x+9y-1) = ______________

The following exercises demonstrate the simplification of problems by using the distributive property followed by combining like terms.  In the process of doing these exercises, you will be learning to show the steps in a neat, well-organized manner.  This will be extremely important in the next section on equation solving.

EXAMPLE 17:            4(x - 3) + 6(4x + 5)

= 4x - 12  +  24x + 30            Distributive Property

                                   = 28x + 18                             Combine like terms

EXAMPLE 18:             6(x - 3) - 4(4x + 5)

                                   = 6x - 18  - 16x – 20              Distributive Property

= -10x - 38                             Combine like terms

EXERCISES:  Remove parentheses and combine like terms.

49.  3(2x + 5) + 6(x + 2)     

    = __________________________     Distributive Property

    = __________________________     Combine like terms

50.  3(2x + 5) - 6(x + 2)     

    = __________________________     Distributive Property

    = __________________________     Combine like terms

51.  3(2x - 5) + 6(x + 2)                               52.      3(6x - 7) + 6(5x - 8)        

     = __________________________                   = ______________________________  

     = __________________________                   = ______________________________

53.  7(3x + 9) - 6(2x + 5)                             54.       5(8x - 4) - 8(5x + 3)        

      = __________________________                 = ______________________________  

     = __________________________                  = ______________________________

55.  3x(5x - 9) + 8(3x - 2)                            56.       2x(6x - 5) + 6(5 - 8x)       

      = __________________________                  = ______________________________  

      = __________________________                  = ______________________________

57.  4x(x + 3) - 8x(5x - 2)                             58.      4x(x + 3) - 8x(5 - 2x)        

     = __________________________                   = _____________________________

     = __________________________                   = ______________________________

59.     9(4x - 5y) - 6(2y - 3x) -  3(6x - 7y)         

       = ______________________________________________________  

       = ______________________________________________________

60.    6x(4x - 5) - 5y(2y + 3) - 4(3x + 8y)         

       = ______________________________________________________  

       = ______________________________________________________

  ANSWERS 1.06

 p. 33 - 38:       

  1. 3x; 2. 2d; 3. 9x; 4. 3b+3g; 5. 2h; 6. 10x; 7. 5c; 8. x;   9. x+3y+4; 10. 8w; 11. 16x-5y;  12. 11k+5v; 13. -6a-c;  14. 13x+5y+z;  15. 3xyz+4xy;   16. 8LH-4WH+2LW;   17. 10x²;   18. 10y²;   19. 7x²+3y²;  20. 7y²+3y; 21.  4x²; 22.  -10y²; 23.  3x² +5x²y²+9y²; 24. 14x²+6x-5y²+6xy; 25. 5x+15; 26. 5x+25; 27. 6x‑24;  28. 8y-72;  29. 14x+35;   30. 20x+35;   31. 54x-48;   32. 63y-81;   33. -5x+25;  34. -6x-30;  35. -3x-7; 36. -3y+4; 37. 8x+7; 38. 8x-5; 39. 5x²-15x;  40. 8y²+72y;  41. 40x²+24xy;   42. 35x²-25xy;  43. 10xy-15x²+35x;  44. 28xy-36y²-24y;  45. -12y²-18xy-42y; 46. -21x²+15xy-3x;   47. -35xy+7x²-7x; 48. 28xy-36y²+4y; 49. 12x+27; 50. 3; 51. 12x-3; 52. 48x-69;  53. 9x+33; 54. -44; 55. 15x²-3x-16;  56. 12x²-58x+30;  57. ‑36x²+28x;  58. 20x²-28x; 59. 36x-36y; 60. 24x²-42x-10y²-47y.